2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 85.0%

Time: 33.7s

Alternatives: 17

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Specification

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{x}^{\left({n}^{-1}\right)}}\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ \mathbf{if}\;n \leq -6900000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, t_1\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_0, -t_0, e^{t_1}\right)\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (sqrt (pow x (pow n -1.0)))) (t_1 (/ (log1p x) n)))
   (if (<= n -6900000.0)
     (-
      (fma 0.5 (/ (pow (log1p x) 2.0) (* n n)) t_1)
      (fma 0.5 (/ (pow (log x) 2.0) (* n n)) (/ (log x) n)))
     (if (<= n 1.2e-7)
       (fma t_0 (- t_0) (exp t_1))
       (if (<= n 3.3e+78)
         (/ (pow x (/ 1.0 n)) (* n x))
         (/ (- (log1p x) (log x)) n))))))

C

double code(double x, double n) {
	double t_0 = sqrt(pow(x, pow(n, -1.0)));
	double t_1 = log1p(x) / n;
	double tmp;
	if (n <= -6900000.0) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), t_1) - fma(0.5, (pow(log(x), 2.0) / (n * n)), (log(x) / n));
	} else if (n <= 1.2e-7) {
		tmp = fma(t_0, -t_0, exp(t_1));
	} else if (n <= 3.3e+78) {
		tmp = pow(x, (1.0 / n)) / (n * x);
	} else {
		tmp = (log1p(x) - log(x)) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = sqrt((x ^ (n ^ -1.0)))
	t_1 = Float64(log1p(x) / n)
	tmp = 0.0
	if (n <= -6900000.0)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), t_1) - fma(0.5, Float64((log(x) ^ 2.0) / Float64(n * n)), Float64(log(x) / n)));
	elseif (n <= 1.2e-7)
		tmp = fma(t_0, Float64(-t_0), exp(t_1));
	elseif (n <= 3.3e+78)
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(n * x));
	else
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Sqrt[N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -6900000.0], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-7], N[(t$95$0 * (-t$95$0) + N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3e+78], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.9e6

   1. Initial program 31.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 79.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 79.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right)} - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      2. log1p-def 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      3. unpow2 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{\color{blue}{n \cdot n}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      4. log1p-def 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      5. fma-def 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{{n}^{2}}, \frac{\log x}{n}\right)} \]

      6. unpow2 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{\color{blue}{n \cdot n}}, \frac{\log x}{n}\right) \]

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)} \]

   if -6.9e6 < n < 1.19999999999999989e-7

   1. Initial program 85.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 85.1%

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)} \]

      2. +-commutative 85.1%

         \[\leadsto \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \]

      3. sqr-pow 85.1%

         \[\leadsto \left(-\color{blue}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \]

      4. distribute-rgt-neg-in 85.1%

         \[\leadsto \color{blue}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \]

      5. fma-def 85.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      6. sqrt-pow1 85.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      7. pow1 85.1%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{1}}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      8. pow1 85.1%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      9. inv-pow 85.1%

         \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\color{blue}{\left({n}^{-1}\right)}}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      10. sqrt-pow1 85.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      11. pow1 85.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{\color{blue}{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{1}}}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      12. pow1 85.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      13. inv-pow 85.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{{x}^{\color{blue}{\left({n}^{-1}\right)}}}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \]

      14. pow-to-exp 85.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{{x}^{\left({n}^{-1}\right)}}, \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}\right) \]

      15. un-div-inv 85.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{{x}^{\left({n}^{-1}\right)}}, e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}\right) \]

      16. +-commutative 85.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{{x}^{\left({n}^{-1}\right)}}, e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}\right) \]

      17. log1p-udef 99.8%

          \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{{x}^{\left({n}^{-1}\right)}}, e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right) \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{{x}^{\left({n}^{-1}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)} \]

   if 1.19999999999999989e-7 < n < 3.3e78

   1. Initial program 16.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 16.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 16.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 16.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 70.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 70.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 70.8%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 70.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 70.8%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 70.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 70.8%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if 3.3e78 < n

   1. Initial program 23.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 83.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 83.0%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 83.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 83.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6900000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{x}^{\left({n}^{-1}\right)}}, -\sqrt{{x}^{\left({n}^{-1}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 2: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ \mathbf{if}\;n \leq -28000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, t_1\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;e^{t_1} - t_0\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log1p x) n)))
   (if (<= n -28000000.0)
     (-
      (fma 0.5 (/ (pow (log1p x) 2.0) (* n n)) t_1)
      (fma 0.5 (/ (pow (log x) 2.0) (* n n)) (/ (log x) n)))
     (if (<= n 1.2e-7)
       (- (exp t_1) t_0)
       (if (<= n 6.6e+77) (/ t_0 (* n x)) (/ (- (log1p x) (log x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log1p(x) / n;
	double tmp;
	if (n <= -28000000.0) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), t_1) - fma(0.5, (pow(log(x), 2.0) / (n * n)), (log(x) / n));
	} else if (n <= 1.2e-7) {
		tmp = exp(t_1) - t_0;
	} else if (n <= 6.6e+77) {
		tmp = t_0 / (n * x);
	} else {
		tmp = (log1p(x) - log(x)) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log1p(x) / n)
	tmp = 0.0
	if (n <= -28000000.0)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), t_1) - fma(0.5, Float64((log(x) ^ 2.0) / Float64(n * n)), Float64(log(x) / n)));
	elseif (n <= 1.2e-7)
		tmp = Float64(exp(t_1) - t_0);
	elseif (n <= 6.6e+77)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -28000000.0], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-7], N[(N[Exp[t$95$1], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[n, 6.6e+77], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.8e7

   1. Initial program 31.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 79.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 79.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right)} - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      2. log1p-def 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      3. unpow2 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{\color{blue}{n \cdot n}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      4. log1p-def 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      5. fma-def 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{{n}^{2}}, \frac{\log x}{n}\right)} \]

      6. unpow2 79.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{\color{blue}{n \cdot n}}, \frac{\log x}{n}\right) \]

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)} \]

   if -2.8e7 < n < 1.19999999999999989e-7

   1. Initial program 85.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around 0 85.1%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. log1p-def 99.8%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1.19999999999999989e-7 < n < 6.5999999999999996e77

   1. Initial program 16.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 16.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 16.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 16.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 70.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 70.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 70.8%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 70.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 70.8%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 70.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 70.8%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if 6.5999999999999996e77 < n

   1. Initial program 23.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 83.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 83.0%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 83.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 83.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -28000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 3: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;\log \left(e^{t_1}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (exp (/ (log1p x) n)) t_0)))
   (if (<= (/ 1.0 n) -0.004)
     (log (exp t_1))
     (if (<= (/ 1.0 n) 5e-84)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 10000000.0) (/ t_0 (* n x)) t_1)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = exp((log1p(x) / n)) - t_0;
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = log(exp(t_1));
	} else if ((1.0 / n) <= 5e-84) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.exp((Math.log1p(x) / n)) - t_0;
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = Math.log(Math.exp(t_1));
	} else if ((1.0 / n) <= 5e-84) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.exp((math.log1p(x) / n)) - t_0
	tmp = 0
	if (1.0 / n) <= -0.004:
		tmp = math.log(math.exp(t_1))
	elif (1.0 / n) <= 5e-84:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 10000000.0:
		tmp = t_0 / (n * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(exp(Float64(log1p(x) / n)) - t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.004)
		tmp = log(exp(t_1));
	elseif (Float64(1.0 / n) <= 5e-84)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 10000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.004], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-84], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -0.0040000000000000001

   1. Initial program 99.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 99.8%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 99.8%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 99.8%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 99.8%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 99.8%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 99.7%

         \[\leadsto \color{blue}{\log \left(e^{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      2. add-log-exp 99.8%

         \[\leadsto \log \left(e^{\color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   if -0.0040000000000000001 < (/.f64 1 n) < 5.0000000000000002e-84

   1. Initial program 27.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 80.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. flip-- 80.8%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}} - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. unpow2 80.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - \color{blue}{{\log x}^{2}}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

   6. Applied egg-rr 80.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]
   7. Step-by-step derivation

      1. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right)} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\log x \cdot \log x}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. flip-- 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}{n} \]

      4. log1p-udef 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      5. diff-log 81.0%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 5.0000000000000002e-84 < (/.f64 1 n) < 1e7

   1. Initial program 16.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 16.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 16.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 16.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 70.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 70.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 70.8%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 70.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 70.8%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 70.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 70.8%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if 1e7 < (/.f64 1 n)

   1. Initial program 52.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around 0 52.1%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. log1p-def 99.9%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -22000000000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= n -22000000000.0)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= n 1.2e-7)
       (- (exp (/ (log1p x) n)) t_0)
       (if (<= n 3.2e+78) (/ t_0 (* n x)) (/ (- (log1p x) (log x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -22000000000.0) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if (n <= 1.2e-7) {
		tmp = exp((log1p(x) / n)) - t_0;
	} else if (n <= 3.2e+78) {
		tmp = t_0 / (n * x);
	} else {
		tmp = (log1p(x) - log(x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (n <= -22000000000.0) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if (n <= 1.2e-7) {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	} else if (n <= 3.2e+78) {
		tmp = t_0 / (n * x);
	} else {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -22000000000.0:
		tmp = math.log(((x + 1.0) / x)) / n
	elif n <= 1.2e-7:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	elif n <= 3.2e+78:
		tmp = t_0 / (n * x)
	else:
		tmp = (math.log1p(x) - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -22000000000.0)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (n <= 1.2e-7)
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	elseif (n <= 3.2e+78)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -22000000000.0], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 1.2e-7], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[n, 3.2e+78], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.2e10

   1. Initial program 31.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 79.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 79.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 79.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 79.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. flip-- 79.1%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

      2. unpow2 79.1%

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}} - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. unpow2 79.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - \color{blue}{{\log x}^{2}}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]
   7. Step-by-step derivation

      1. unpow2 79.1%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right)} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      2. unpow2 79.1%

         \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\log x \cdot \log x}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. flip-- 79.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}{n} \]

      4. log1p-udef 79.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      5. diff-log 79.3%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   8. Applied egg-rr 79.3%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if -2.2e10 < n < 1.19999999999999989e-7

   1. Initial program 85.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around 0 85.1%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. log1p-def 99.8%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1.19999999999999989e-7 < n < 3.19999999999999994e78

   1. Initial program 16.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 16.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 16.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 16.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 70.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 70.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 70.8%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 70.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 70.8%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 70.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 70.8%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if 3.19999999999999994e78 < n

   1. Initial program 23.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 83.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 83.0%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 83.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 83.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -22000000000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]

Alternative 5: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.004)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 5e-84)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 10000000.0)
         (/ t_0 (* n x))
         (-
          (+ 1.0 (fma (* x x) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ x n)))
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = (1.0 + fma((x * x), ((0.5 / (n * n)) - (0.5 / n)), (x / n))) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.004)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 5e-84)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 10000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(Float64(1.0 + fma(Float64(x * x), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(x / n))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.004], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-84], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -0.0040000000000000001

   1. Initial program 99.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if -0.0040000000000000001 < (/.f64 1 n) < 5.0000000000000002e-84

   1. Initial program 27.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 80.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. flip-- 80.8%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}} - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. unpow2 80.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - \color{blue}{{\log x}^{2}}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

   6. Applied egg-rr 80.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]
   7. Step-by-step derivation

      1. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right)} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\log x \cdot \log x}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. flip-- 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}{n} \]

      4. log1p-udef 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      5. diff-log 81.0%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 5.0000000000000002e-84 < (/.f64 1 n) < 1e7

   1. Initial program 16.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 16.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 16.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 16.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 70.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 70.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 70.8%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 70.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 70.8%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 70.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 70.8%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if 1e7 < (/.f64 1 n)

   1. Initial program 52.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 70.4%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 70.4%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 70.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 70.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 70.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 70.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 70.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 70.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 70.4%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= (/ 1.0 n) -0.004)
     t_1
     (if (<= (/ 1.0 n) 5e-84)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 10000000.0)
         (/ t_0 (* n x))
         (if (<= (/ 1.0 n) 1e+143) t_1 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e+143) {
		tmp = t_1;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if ((1.0d0 / n) <= (-0.004d0)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-84) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 10000000.0d0) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d+143) then
        tmp = t_1
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e+143) {
		tmp = t_1;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if (1.0 / n) <= -0.004:
		tmp = t_1
	elif (1.0 / n) <= 5e-84:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 10000000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e+143:
		tmp = t_1
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.004)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-84)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 10000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e+143)
		tmp = t_1;
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if ((1.0 / n) <= -0.004)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-84)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 10000000.0)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e+143)
		tmp = t_1;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.004], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-84], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+143], t$95$1, N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -0.0040000000000000001 or 1e7 < (/.f64 1 n) < 1e143

   1. Initial program 94.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if -0.0040000000000000001 < (/.f64 1 n) < 5.0000000000000002e-84

   1. Initial program 27.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 80.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. flip-- 80.8%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}} - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. unpow2 80.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - \color{blue}{{\log x}^{2}}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

   6. Applied egg-rr 80.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]
   7. Step-by-step derivation

      1. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right)} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\log x \cdot \log x}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. flip-- 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}{n} \]

      4. log1p-udef 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      5. diff-log 81.0%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 5.0000000000000002e-84 < (/.f64 1 n) < 1e7

   1. Initial program 16.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 16.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 16.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 16.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 16.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 70.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 70.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 70.8%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 70.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 70.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 70.8%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 70.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 70.8%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if 1e143 < (/.f64 1 n)

   1. Initial program 31.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 6.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 6.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 6.1%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 81.8%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 81.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 81.8%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 81.8%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 7: 71.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -5e+81)
     (/ (/ 0.3333333333333333 (pow x 3.0)) n)
     (if (<= (/ 1.0 n) -5e+19)
       t_0
       (if (<= (/ 1.0 n) 5e-84)
         (/ (log (/ (+ x 1.0) x)) n)
         (if (<= (/ 1.0 n) 2e-21)
           (/ (/ 1.0 n) x)
           (if (<= (/ 1.0 n) 1e+143) t_0 (/ (/ n x) (* n n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+81) {
		tmp = (0.3333333333333333 / pow(x, 3.0)) / n;
	} else if ((1.0 / n) <= -5e+19) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-21) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 1e+143) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-5d+81)) then
        tmp = (0.3333333333333333d0 / (x ** 3.0d0)) / n
    else if ((1.0d0 / n) <= (-5d+19)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-84) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d-21) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 1d+143) then
        tmp = t_0
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+81) {
		tmp = (0.3333333333333333 / Math.pow(x, 3.0)) / n;
	} else if ((1.0 / n) <= -5e+19) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-21) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 1e+143) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e+81:
		tmp = (0.3333333333333333 / math.pow(x, 3.0)) / n
	elif (1.0 / n) <= -5e+19:
		tmp = t_0
	elif (1.0 / n) <= 5e-84:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e-21:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 1e+143:
		tmp = t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+81)
		tmp = Float64(Float64(0.3333333333333333 / (x ^ 3.0)) / n);
	elseif (Float64(1.0 / n) <= -5e+19)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-84)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-21)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e+143)
		tmp = t_0;
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -5e+81)
		tmp = (0.3333333333333333 / (x ^ 3.0)) / n;
	elseif ((1.0 / n) <= -5e+19)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-84)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e-21)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 1e+143)
		tmp = t_0;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+81], N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+19], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-84], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-21], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+143], t$95$0, N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.9999999999999998e81

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 53.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 53.0%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 53.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 53.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 23.8%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]

   6. Taylor expanded in x around 0 80.8%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}}}}{n} \]

   if -4.9999999999999998e81 < (/.f64 1 n) < -5e19 or 1.99999999999999982e-21 < (/.f64 1 n) < 1e143

   1. Initial program 79.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 65.9%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if -5e19 < (/.f64 1 n) < 5.0000000000000002e-84

   1. Initial program 31.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 81.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 81.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 81.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 81.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. flip-- 81.4%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

      2. unpow2 81.4%

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}} - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. unpow2 81.4%

         \[\leadsto \frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - \color{blue}{{\log x}^{2}}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

   6. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]
   7. Step-by-step derivation

      1. unpow2 81.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right)} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      2. unpow2 81.4%

         \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\log x \cdot \log x}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. flip-- 81.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}{n} \]

      4. log1p-udef 81.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      5. diff-log 81.5%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   8. Applied egg-rr 81.5%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 5.0000000000000002e-84 < (/.f64 1 n) < 1.99999999999999982e-21

   1. Initial program 12.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 31.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 31.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right)} - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      2. log1p-def 31.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      3. unpow2 31.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{\color{blue}{n \cdot n}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      4. log1p-def 31.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      5. fma-def 31.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{{n}^{2}}, \frac{\log x}{n}\right)} \]

      6. unpow2 31.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{\color{blue}{n \cdot n}}, \frac{\log x}{n}\right) \]

   4. Simplified 31.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)} \]

   5. Taylor expanded in x around inf 80.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}} + \frac{1}{n}}{x}} \]
   6. Step-by-step derivation

      1. +-commutative 80.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{n} + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}}}{x} \]

      2. mul-1-neg 80.7%

         \[\leadsto \frac{\frac{1}{n} + \color{blue}{\left(-\frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}}{x} \]

      3. distribute-frac-neg 80.7%

         \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{-\log \left(\frac{1}{x}\right)}{{n}^{2}}}}{x} \]

      4. log-rec 80.7%

         \[\leadsto \frac{\frac{1}{n} + \frac{-\color{blue}{\left(-\log x\right)}}{{n}^{2}}}{x} \]

      5. remove-double-neg 80.7%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

      6. unpow2 80.7%

         \[\leadsto \frac{\frac{1}{n} + \frac{\log x}{\color{blue}{n \cdot n}}}{x} \]

   7. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}} \]

   8. Taylor expanded in n around inf 80.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   if 1e143 < (/.f64 1 n)

   1. Initial program 31.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 6.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 6.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 6.1%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 81.8%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 81.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 81.8%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 81.8%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+19}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 8: 80.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -0.004)
     t_1
     (if (<= (/ 1.0 n) 5e-84)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 10000000.0)
         t_1
         (if (<= (/ 1.0 n) 1e+143)
           (- (+ 1.0 (/ x n)) t_0)
           (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+143) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-0.004d0)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-84) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 10000000.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+143) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+143) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -0.004:
		tmp = t_1
	elif (1.0 / n) <= 5e-84:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 10000000.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+143:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.004)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-84)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 10000000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+143)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -0.004)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-84)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 10000000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+143)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.004], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-84], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10000000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+143], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -0.0040000000000000001 or 5.0000000000000002e-84 < (/.f64 1 n) < 1e7

   1. Initial program 86.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 86.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 86.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 86.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 86.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 86.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 94.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 94.5%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 94.5%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 94.5%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 94.5%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 94.5%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 94.5%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 94.5%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 94.5%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 94.5%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -0.0040000000000000001 < (/.f64 1 n) < 5.0000000000000002e-84

   1. Initial program 27.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 80.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. flip-- 80.8%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}} - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. unpow2 80.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - \color{blue}{{\log x}^{2}}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

   6. Applied egg-rr 80.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]
   7. Step-by-step derivation

      1. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right)} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\log x \cdot \log x}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. flip-- 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}{n} \]

      4. log1p-udef 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      5. diff-log 81.0%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 1e7 < (/.f64 1 n) < 1e143

   1. Initial program 71.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1e143 < (/.f64 1 n)

   1. Initial program 31.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 6.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 6.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 6.1%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 81.8%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 81.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 81.8%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 81.8%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 9: 80.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -0.004)
     t_1
     (if (<= (/ 1.0 n) 5e-84)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 10000000.0)
         t_1
         (if (<= (/ 1.0 n) 1e+143) (- 1.0 t_0) (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+143) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-0.004d0)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-84) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 10000000.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+143) then
        tmp = 1.0d0 - t_0
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -0.004) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-84) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 10000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+143) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -0.004:
		tmp = t_1
	elif (1.0 / n) <= 5e-84:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 10000000.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+143:
		tmp = 1.0 - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.004)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-84)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 10000000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+143)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -0.004)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-84)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 10000000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+143)
		tmp = 1.0 - t_0;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.004], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-84], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10000000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+143], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -0.0040000000000000001 or 5.0000000000000002e-84 < (/.f64 1 n) < 1e7

   1. Initial program 86.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-log-exp 86.5%

         \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow-to-exp 86.5%

         \[\leadsto \log \left(e^{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv 86.5%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. +-commutative 86.5%

         \[\leadsto \log \left(e^{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. log1p-udef 86.5%

         \[\leadsto \log \left(e^{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around inf 94.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   5. Step-by-step derivation

      1. mul-1-neg 94.5%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 94.5%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-neg-frac 94.5%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 94.5%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-lft-identity 94.5%

         \[\leadsto \frac{e^{\frac{\color{blue}{1 \cdot \log x}}{n}}}{n \cdot x} \]

      6. associate-*l/ 94.5%

         \[\leadsto \frac{e^{\color{blue}{\frac{1}{n} \cdot \log x}}}{n \cdot x} \]

      7. *-commutative 94.5%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      8. exp-to-pow 94.5%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      9. *-commutative 94.5%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   6. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   if -0.0040000000000000001 < (/.f64 1 n) < 5.0000000000000002e-84

   1. Initial program 27.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 80.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. flip-- 80.8%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}} - \log x \cdot \log x}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. unpow2 80.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - \color{blue}{{\log x}^{2}}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

   6. Applied egg-rr 80.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}}{n} \]
   7. Step-by-step derivation

      1. unpow2 80.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right)} - {\log x}^{2}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      2. unpow2 80.8%

         \[\leadsto \frac{\frac{\mathsf{log1p}\left(x\right) \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\log x \cdot \log x}}{\mathsf{log1p}\left(x\right) + \log x}}{n} \]

      3. flip-- 80.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}{n} \]

      4. log1p-udef 80.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      5. diff-log 81.0%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 1e7 < (/.f64 1 n) < 1e143

   1. Initial program 71.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 65.9%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1e143 < (/.f64 1 n)

   1. Initial program 31.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 6.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 6.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 6.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 6.1%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 81.8%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 81.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 81.8%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 81.8%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 81.8%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.004:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 10: 59.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := \frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \mathbf{if}\;x \leq 1.4 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-230}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)) (t_1 (/ (/ 0.3333333333333333 (pow x 3.0)) n)))
   (if (<= x 1.4e-259)
     t_0
     (if (<= x 7.5e-230)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= x 3.3e-97)
         t_0
         (if (<= x 7.5e-74)
           t_1
           (if (<= x 0.98)
             (/ (- x (log x)) n)
             (if (<= x 1.15e+113)
               (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
               t_1))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double t_1 = (0.3333333333333333 / pow(x, 3.0)) / n;
	double tmp;
	if (x <= 1.4e-259) {
		tmp = t_0;
	} else if (x <= 7.5e-230) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 3.3e-97) {
		tmp = t_0;
	} else if (x <= 7.5e-74) {
		tmp = t_1;
	} else if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.15e+113) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -log(x) / n
    t_1 = (0.3333333333333333d0 / (x ** 3.0d0)) / n
    if (x <= 1.4d-259) then
        tmp = t_0
    else if (x <= 7.5d-230) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 3.3d-97) then
        tmp = t_0
    else if (x <= 7.5d-74) then
        tmp = t_1
    else if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.15d+113) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double t_1 = (0.3333333333333333 / Math.pow(x, 3.0)) / n;
	double tmp;
	if (x <= 1.4e-259) {
		tmp = t_0;
	} else if (x <= 7.5e-230) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 3.3e-97) {
		tmp = t_0;
	} else if (x <= 7.5e-74) {
		tmp = t_1;
	} else if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.15e+113) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	t_1 = (0.3333333333333333 / math.pow(x, 3.0)) / n
	tmp = 0
	if x <= 1.4e-259:
		tmp = t_0
	elif x <= 7.5e-230:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 3.3e-97:
		tmp = t_0
	elif x <= 7.5e-74:
		tmp = t_1
	elif x <= 0.98:
		tmp = (x - math.log(x)) / n
	elif x <= 1.15e+113:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	t_1 = Float64(Float64(0.3333333333333333 / (x ^ 3.0)) / n)
	tmp = 0.0
	if (x <= 1.4e-259)
		tmp = t_0;
	elseif (x <= 7.5e-230)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 3.3e-97)
		tmp = t_0;
	elseif (x <= 7.5e-74)
		tmp = t_1;
	elseif (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.15e+113)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	t_1 = (0.3333333333333333 / (x ^ 3.0)) / n;
	tmp = 0.0;
	if (x <= 1.4e-259)
		tmp = t_0;
	elseif (x <= 7.5e-230)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 3.3e-97)
		tmp = t_0;
	elseif (x <= 7.5e-74)
		tmp = t_1;
	elseif (x <= 0.98)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.15e+113)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 1.4e-259], t$95$0, If[LessEqual[x, 7.5e-230], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-97], t$95$0, If[LessEqual[x, 7.5e-74], t$95$1, If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.15e+113], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.4e-259 or 7.50000000000000006e-230 < x < 3.3000000000000001e-97

   1. Initial program 33.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 67.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 67.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 67.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 67.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 67.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 67.9%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 67.9%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 1.4e-259 < x < 7.50000000000000006e-230

   1. Initial program 76.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 76.4%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 3.3000000000000001e-97 < x < 7.5e-74 or 1.14999999999999998e113 < x

   1. Initial program 75.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 69.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 69.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 69.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 69.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 69.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.5%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]

   6. Taylor expanded in x around 0 79.4%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}}}}{n} \]

   if 7.5e-74 < x < 0.97999999999999998

   1. Initial program 41.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 54.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 54.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 54.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 54.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 52.4%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 52.4%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 52.4%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 52.4%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.97999999999999998 < x < 1.14999999999999998e113

   1. Initial program 37.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 39.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 39.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 39.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 39.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 39.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.8%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.8%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.8%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-259}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-230}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \end{array} \]

Alternative 11: 56.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-230}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 1.9e-258)
     t_0
     (if (<= x 6e-230)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= x 7.6e-90)
         t_0
         (if (<= x 1.1e-73)
           (/ (/ n x) (* n n))
           (if (<= x 1.0)
             (/ (- x (log x)) n)
             (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 1.9e-258) {
		tmp = t_0;
	} else if (x <= 6e-230) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 7.6e-90) {
		tmp = t_0;
	} else if (x <= 1.1e-73) {
		tmp = (n / x) / (n * n);
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 1.9d-258) then
        tmp = t_0
    else if (x <= 6d-230) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 7.6d-90) then
        tmp = t_0
    else if (x <= 1.1d-73) then
        tmp = (n / x) / (n * n)
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 1.9e-258) {
		tmp = t_0;
	} else if (x <= 6e-230) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 7.6e-90) {
		tmp = t_0;
	} else if (x <= 1.1e-73) {
		tmp = (n / x) / (n * n);
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 1.9e-258:
		tmp = t_0
	elif x <= 6e-230:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 7.6e-90:
		tmp = t_0
	elif x <= 1.1e-73:
		tmp = (n / x) / (n * n)
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 1.9e-258)
		tmp = t_0;
	elseif (x <= 6e-230)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 7.6e-90)
		tmp = t_0;
	elseif (x <= 1.1e-73)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 1.9e-258)
		tmp = t_0;
	elseif (x <= 6e-230)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 7.6e-90)
		tmp = t_0;
	elseif (x <= 1.1e-73)
		tmp = (n / x) / (n * n);
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 1.9e-258], t$95$0, If[LessEqual[x, 6e-230], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.6e-90], t$95$0, If[LessEqual[x, 1.1e-73], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.8999999999999999e-258 or 6e-230 < x < 7.6e-90

   1. Initial program 33.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 66.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 66.1%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 66.1%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 1.8999999999999999e-258 < x < 6e-230

   1. Initial program 76.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 76.4%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 7.6e-90 < x < 1.1e-73

   1. Initial program 51.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 12.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 12.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 12.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 12.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 12.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 12.9%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 72.8%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 72.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 72.8%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 72.8%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 72.8%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 66.2%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]

   if 1.1e-73 < x < 1

   1. Initial program 41.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 54.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 54.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 54.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 54.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 52.4%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 52.4%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 52.4%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 52.4%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 1 < x

   1. Initial program 65.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-230}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 12: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.6e-90)
   (/ (- (log x)) n)
   (if (<= x 7.5e-74)
     (/ (/ n x) (* n n))
     (if (<= x 0.95)
       (/ (- x (log x)) n)
       (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.6e-90) {
		tmp = -log(x) / n;
	} else if (x <= 7.5e-74) {
		tmp = (n / x) / (n * n);
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7.6d-90) then
        tmp = -log(x) / n
    else if (x <= 7.5d-74) then
        tmp = (n / x) / (n * n)
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7.6e-90) {
		tmp = -Math.log(x) / n;
	} else if (x <= 7.5e-74) {
		tmp = (n / x) / (n * n);
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.6e-90:
		tmp = -math.log(x) / n
	elif x <= 7.5e-74:
		tmp = (n / x) / (n * n)
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.6e-90)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 7.5e-74)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7.6e-90)
		tmp = -log(x) / n;
	elseif (x <= 7.5e-74)
		tmp = (n / x) / (n * n);
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.6e-90], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 7.5e-74], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.6e-90

   1. Initial program 38.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 61.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 61.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 61.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 61.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 61.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 61.5%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 61.5%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 7.6e-90 < x < 7.5e-74

   1. Initial program 51.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 12.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 12.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 12.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 12.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 12.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 12.9%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 72.8%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 72.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 72.8%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 72.8%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 72.8%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 66.2%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]

   if 7.5e-74 < x < 0.94999999999999996

   1. Initial program 41.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 54.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 54.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 54.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 54.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 52.4%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 52.4%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 52.4%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 52.4%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.94999999999999996 < x

   1. Initial program 65.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 13: 56.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 6.5e-90)
     t_0
     (if (<= x 7.5e-74)
       (/ (/ n x) (* n n))
       (if (<= x 0.7) t_0 (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 6.5e-90) {
		tmp = t_0;
	} else if (x <= 7.5e-74) {
		tmp = (n / x) / (n * n);
	} else if (x <= 0.7) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 6.5d-90) then
        tmp = t_0
    else if (x <= 7.5d-74) then
        tmp = (n / x) / (n * n)
    else if (x <= 0.7d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 6.5e-90) {
		tmp = t_0;
	} else if (x <= 7.5e-74) {
		tmp = (n / x) / (n * n);
	} else if (x <= 0.7) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 6.5e-90:
		tmp = t_0
	elif x <= 7.5e-74:
		tmp = (n / x) / (n * n)
	elif x <= 0.7:
		tmp = t_0
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 6.5e-90)
		tmp = t_0;
	elseif (x <= 7.5e-74)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif (x <= 0.7)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 6.5e-90)
		tmp = t_0;
	elseif (x <= 7.5e-74)
		tmp = (n / x) / (n * n);
	elseif (x <= 0.7)
		tmp = t_0;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 6.5e-90], t$95$0, If[LessEqual[x, 7.5e-74], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.7], t$95$0, N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.4999999999999996e-90 or 7.5e-74 < x < 0.69999999999999996

   1. Initial program 39.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 59.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 59.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 59.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 59.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 58.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 58.6%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 58.6%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 6.4999999999999996e-90 < x < 7.5e-74

   1. Initial program 51.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 12.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 12.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 12.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 12.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 12.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 12.9%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 72.8%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 72.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 72.8%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 72.8%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 72.8%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 66.2%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]

   if 0.69999999999999996 < x

   1. Initial program 65.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 14: 44.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0) (/ (/ n x) (* n n)) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (n / x) / (n * n)
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (n / x) / (n * n)
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (n / x) / (n * n);
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 40.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 55.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 55.0%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 55.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 55.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 55.0%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 49.2%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 49.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 49.2%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 49.2%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 49.2%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 49.2%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 28.3%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]

   if 1 < x

   1. Initial program 65.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.7%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 15: 43.9% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0065:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0065) (/ (/ n x) (* n n)) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0065) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0065d0) then
        tmp = (n / x) / (n * n)
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0065) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0065:
		tmp = (n / x) / (n * n)
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0065)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0065)
		tmp = (n / x) / (n * n);
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0065], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0064999999999999997

   1. Initial program 40.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 55.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 55.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 55.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 55.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 55.1%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. frac-sub 49.2%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]

   8. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot n - n \cdot \log x}{n \cdot n}} \]
   9. Step-by-step derivation

      1. log1p-def 49.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} \cdot n - n \cdot \log x}{n \cdot n} \]

      2. *-commutative 49.2%

         \[\leadsto \frac{\color{blue}{n \cdot \log \left(1 + x\right)} - n \cdot \log x}{n \cdot n} \]

      3. distribute-lft-out-- 49.2%

         \[\leadsto \frac{\color{blue}{n \cdot \left(\log \left(1 + x\right) - \log x\right)}}{n \cdot n} \]

      4. log1p-def 49.2%

         \[\leadsto \frac{n \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log x\right)}{n \cdot n} \]

   10. Simplified 49.2%

       \[\leadsto \color{blue}{\frac{n \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n \cdot n}} \]

   11. Taylor expanded in x around inf 28.6%

       \[\leadsto \frac{\color{blue}{\frac{n}{x}}}{n \cdot n} \]

   if 0.0064999999999999997 < x

   1. Initial program 65.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 50.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 50.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right)} - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      2. log1p-def 50.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      3. unpow2 50.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{\color{blue}{n \cdot n}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      4. log1p-def 50.7%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      5. fma-def 50.7%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{{n}^{2}}, \frac{\log x}{n}\right)} \]

      6. unpow2 50.7%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{\color{blue}{n \cdot n}}, \frac{\log x}{n}\right) \]

   4. Simplified 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)} \]

   5. Taylor expanded in x around inf 62.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}} + \frac{1}{n}}{x}} \]
   6. Step-by-step derivation

      1. +-commutative 62.0%

         \[\leadsto \frac{\color{blue}{\frac{1}{n} + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}}}{x} \]

      2. mul-1-neg 62.0%

         \[\leadsto \frac{\frac{1}{n} + \color{blue}{\left(-\frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}}{x} \]

      3. distribute-frac-neg 62.0%

         \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{-\log \left(\frac{1}{x}\right)}{{n}^{2}}}}{x} \]

      4. log-rec 62.0%

         \[\leadsto \frac{\frac{1}{n} + \frac{-\color{blue}{\left(-\log x\right)}}{{n}^{2}}}{x} \]

      5. remove-double-neg 62.0%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

      6. unpow2 62.0%

         \[\leadsto \frac{\frac{1}{n} + \frac{\log x}{\color{blue}{n \cdot n}}}{x} \]

   7. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}} \]

   8. Taylor expanded in n around inf 62.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0065:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 16: 41.1% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf 59.8%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. +-rgt-identity 59.8%

      \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

   2. +-rgt-identity 59.8%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

   3. log1p-def 59.8%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

4. Simplified 59.8%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf 38.8%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
6. Step-by-step derivation

   1. *-commutative 38.8%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

7. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

8. Final simplification 38.8%

   \[\leadsto \frac{1}{n \cdot x} \]

Alternative 17: 41.6% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf 52.6%

   \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)} \]
3. Step-by-step derivation

   1. fma-def 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right)} - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

   2. log1p-def 52.6%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

   3. unpow2 52.6%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{\color{blue}{n \cdot n}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

   4. log1p-def 52.6%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

   5. fma-def 52.6%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{{n}^{2}}, \frac{\log x}{n}\right)} \]

   6. unpow2 52.6%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{\color{blue}{n \cdot n}}, \frac{\log x}{n}\right) \]

4. Simplified 52.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)} \]

5. Taylor expanded in x around inf 39.6%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}} + \frac{1}{n}}{x}} \]
6. Step-by-step derivation

   1. +-commutative 39.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{n} + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}}}{x} \]

   2. mul-1-neg 39.6%

      \[\leadsto \frac{\frac{1}{n} + \color{blue}{\left(-\frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}}{x} \]

   3. distribute-frac-neg 39.6%

      \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{-\log \left(\frac{1}{x}\right)}{{n}^{2}}}}{x} \]

   4. log-rec 39.6%

      \[\leadsto \frac{\frac{1}{n} + \frac{-\color{blue}{\left(-\log x\right)}}{{n}^{2}}}{x} \]

   5. remove-double-neg 39.6%

      \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

   6. unpow2 39.6%

      \[\leadsto \frac{\frac{1}{n} + \frac{\log x}{\color{blue}{n \cdot n}}}{x} \]

7. Simplified 39.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}} \]

8. Taylor expanded in n around inf 39.3%

   \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

9. Final simplification 39.3%

   \[\leadsto \frac{\frac{1}{n}}{x} \]

Reproduce

herbie shell --seed 2023280 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))